$2.5 + 5.9 + 8.13 + 11.17 + \ldots$ to $10$ terms $=$

Let $\langle a_n \rangle$ be a sequence such that $a_1+a_2+\ldots+a_n = \frac{n^2+3n}{(n+1)(n+2)}$. If $28 \sum_{k=1}^{10} \frac{1}{a_k} = p_1 p_2 p_3 \ldots p_m$,where $p_1, p_2, \ldots, p_m$ are the first $m$ prime numbers,then $m$ is equal to

Let $a_{n}$ be the $n^{\text{th}}$ term of the series $5+8+14+23+35+50+\ldots$ and $S_{n}=\sum_{k=1}^{n} a_{k}$. Then $S_{30}-a_{40}$ is equal to

The number of solutions of $2^{1+|\cos x|+|\cos x|^2+\ldots} = 4$ in $(-\pi, \pi)$ is

If $1+(1-2^{2} \cdot 1)+(1-4^{2} \cdot 3)+(1-6^{2} \cdot 5)+\ldots+(1-20^{2} \cdot 19) = \alpha - 220 \beta$,then the ordered pair $(\alpha, \beta)$ is equal to:

If $2^3+4^3+6^3+\ldots+(2n)^3=h n^2(n+1)^2$,then $h$ is equal to